In [1]}, Petersen and Wei generalized the classical Bishop-Gromov volume comparison in an integral bound for the Ricci tensor. Before recalling their results, we need some notation. On a Riemannian manifold \(M\) , let \(Ric_{-}\)  be the smallest eigenvalue for the Ricci tensor \(Ric: T_{x}M \rightarrow T_{x}M\) 
and
\(Ric^{\, H}_{-}:=\left((n-1)H - Ric_{-}\right)_{+} = \mbox{\rm max}\lbrace 0, (n-1)H - Ric_{-}\rbrace \) 
